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"""
Euler discovered the remarkable quadratic formula:
n² + n + 41
It turns out that the formula will produce 40 primes for the consecutive values n = 0 to 39.
However, when n = 40, 402 + 40 + 41 = 40(40 + 1) + 41 is divisible by 41,
and certainly when n = 41, 41² + 41 + 41 is clearly divisible by 41.
The incredible formula n² − 79n + 1601 was discovered, which produces 80 primes for the
consecutive values n = 0 to 79. The product of the coefficients, −79 and 1601, is −126479.
Considering quadratics of the form:
n² + an + b, where |a| < 1000 and |b| < 1000
where |n| is the modulus/absolute value of n
e.g. |11| = 11 and |−4| = 4
Find the product of the coefficients, a and b, for the quadratic expression that produces the
maximum number of primes for consecutive values of n, starting with n = 0.
"""
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import itertools
from sympy.ntheory import isprime
limit = 1000
abcombos = list(itertools.product(range(-(limit - 1),limit), repeat = 2))
n= 0
maxab = (0,0)
maxn = 0
for a, b in abcombos:
n = 0 # start with n = 0
while isprime((n+1)**2 + a*(n+1) + b): # while quadratic expression per (a,b) tuple produces primes:
n += 1 # increment result counter
if n > maxn: # when the expression produces a non-prime for the first time, check if the previous run
maxn = n # new n with the most primes per (a,b) tuple
maxab = a, b # new (a,b) tuple
print(maxab[0]*maxab[1])
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